However, despite its interesting properties, and in contrast with VaR, blacklittle work exists on modeling ES. This is in part due to the fact that ES is not an “elicitable" measure, in the sense that there does not exist a loss function such that ES is the solution that minimizes the expected loss. Several works have been proposed in the literature to overcome the problem of elicitability (see, e.g., [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}). Recently, using the results of [8]}, who show that ES is jointly elicitable with VaR, [9]} uses the Asymmetric Laplace (AL) distribution to jointly estimate dynamic models for both VaR and ES. In particular, [9]} shows that the negative of the log-likelihood associated to the AL distribution belongs to the class of loss functions presented in [8]}, and hence it can be used to estimate and forecast VaR and ES measures in one step. In his paper, the joint estimation of VaR and ES is obtained blackin a univariate quantile regression framework, blackexploiting
the interesting result that ES can be expressed in terms of the scale parameter of the AL density.
